Compared to speedrunning other games, Minecraft Speedrunning is very random. In the catagory of Random Seed Glitchless, players have to react to a completely new seed with no prior information and make decisions, most of the time the decisions do not have deterministic results and lead them to different paths.
Something interesting happens when a game like this is made competitive into what’s known as ranked, where players are given the same seeds filtered to be reasonably good(that they still have little to no prior information to) and compete in different worlds to see who get a faster time.
The Math Fact
Suppose we have 3 random variables X_1, X_2\ \text{and}\ X_3. Then P(X_1 < X_2) > \frac{1}{2} and P(X_2 < X_3) > \frac{1}{2} does not imply P(X_1 < X_3) > \frac{1}{2}.
Constructive Proof:
Let a = \frac{\sqrt{5} - 1}{2} \approx 0.61803 > \frac{1}{2}. Note that 1 - a = a^2.
- P(X_1 = 1) = 1-a, P(X_1 = 4) = a;
- P(X_2 = 2) = a, P(X_2 = 5) = 1-a;
- P(X_3 = 3) = 1.
- X_1, X_2\ \text{and}\ X_3 are independent.
Then:
- P(X_1 < X_2) = P(X_1 = 1) + P(X_1 = 4) \times P(X_2 = 5) =
(1-a) + a(1-a) = 1 - a^2 = a > \frac{1}{2}. - P(X_2 < X_3) = P(X_2 = 2) = a > \frac{1}{2}.
- P(X_1 < X_3) = P(X_1 = 1) = 1-a < \frac{1}{2}.
In fact, this has been proven to be the most extreme 3 variable case, where the minimum of P(X_1 < X_2), P(X_2 < X_3) and P(X_3 < X_1) max out at \frac{\sqrt{5} - 1}{2} \approx 0.61803 in Stanisław Trybuła, "On the paradox of n random variables", Zastosowania Matematyki (1965). With more and more players, \lim\limits_{n \to \infty}\text{sup}\ \text{min}(P(X_1 < X_2), P(X_2 < X_3), \dots , P(X_n < X_1)) = \frac{3}{4}.
What does it tell us?
Think about the random variables as the time a player gets, and when a player gets a faster time than the other they win. Since Ranked games have so much internal and external randomness, it allows for such different play styles, meaning that a case that P(\text{A beats B}) > \frac{1}{2}, P(\text{B beats C}) > \frac{1}{2}, P(\text{C beats A}) > \frac{1}{2} can and most definitely happened. With such players, any rating system, whether it is Elo or Glicko or whatever modifications you made based on them, so long as it follows win rate transitivity, it will collapse, and your rating will not converge, instead it will be changing depending on the people that you queue. Trying to make a model with a single number as a player’s rating to predict their 1v1 win rate against each other will never work.
The reason why this is rarely mentioned in other games is usually the result of a certain decision yields a relatively determinisic results, and both players play in different worlds so they are much more similar to independent random variables. Some games like chess for example, is so deterministic that the rating system fail to predict the win rate of a high level player to an average player that they can farm rating infinitely without having to worry about ever losing a game; Whereas in ranked, a top player can always just die in lava and its not your fault at all while still losing to a 1200 that is playing completely normally.
One other thing I want to mention is tournaments. While a tournament setting can definitely benifit certain players (for example, an imaginary player that forfeits 60% of the time but finishes sub 7 otherwise can be ranked coal on the ladder, but still perform better than a few players in playoffs), and the seeds are suspected to have different standards than normal ladder seeds which allows for different play styles, in every game the only thing that matters is still a player’s win rate in that single game, and your only objective as a player is to maximize that win rate in that single game. For example, some people believe that playing risky in a playoffs setting is more benificial, because ‘you only have to win those few games’. That is in fact not true, when you do make a risky play in a game, you change your chance of winning that single game, and if you do get unlucky, you do not get karma points that transfer to the next game that makes your next risky play more likely to pay out.